# Writing del, divergence, and curl in generalized coordinates

In three dimensional Cartesian coordinates the Hamilton operator, del, is written as

$$\nabla= \begin{pmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{pmatrix}$$

The divergence of a vector field $$A$$ is written as

$$\nabla \cdot A= \begin{pmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{pmatrix} \cdot \begin{pmatrix} A_x \\ A_y \\ A_z \end{pmatrix} =\frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}$$

The curl is writtten as

$$\nabla \times A= \begin{pmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{pmatrix} \times \begin{pmatrix} A_x \\ A_y \\ A_z \end{pmatrix} = \begin{pmatrix} \frac{\partial A_z}{\partial y} -\frac{\partial A_y}{\partial z} \\ \frac{\partial A_x}{\partial z} -\frac{\partial A_z}{\partial x} \\ \frac{\partial A_y}{\partial x} -\frac{\partial A_x}{\partial y} \end{pmatrix}$$

I am trying to write these three equations for generalized coordinates using the Einstein summation convention and the basis vectors $$e_1$$, $$e_2$$, and $$e_3$$ for the scalars $$x^1$$, $$x^2$$, and $$x^3$$

So far I have the equations for gradient and divergence as

$$\nabla= e_\mu \frac{\partial}{\partial x^\mu}$$

$$\nabla \cdot A= e_\mu \cdot\frac{\partial A_\mu}{\partial x^\mu}$$

Are these equations correct? What would the equation for curl be?

• @DomDoe that is the Hamiltonian operator – Ryan Parikh Oct 29 at 8:48
• No, they're not correct at all (and the second one doesn't make sense to begin with). There's a Wikipedia page describing the formulas for typical specific cases and one with the general case. Why don't you start there, and come back here if you have specific questions? – E.P. Oct 29 at 8:53
• The specific pages for grad (en.wikipedia.org/wiki/Gradient), div (en.wikipedia.org/wiki/Divergence), curl (en.wikipedia.org/wiki/Curl_(mathematics)) also have generalized versions. You will need some grasp of what vectors and co-vectors are, what is metric, what is covariant derivative, what is the connection, Levi-Civita relative tensors, and generalized Kroenecker deltas. This is for completely arbitrary coordinates. Simpler expressions are available if you stick with curvilinear coordinates – Cryo Oct 29 at 11:37
• @Cryo "You will need some grasp of [long list of things that are arcane and mysterious right up until they are easy and obvious]" OK. That made me chuckle. – dmckee Oct 29 at 15:08